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Journal of Economic Theory 147 (2012) 1460–1489
www.elsevier.com/locate/jet

Increasing interdependence of multivariate
distributions ✩Margaret Meyer a,∗ , Bruno Strulovici b
a Nuffield College and Dept. of Economics, Oxford University, United Kingdomb Dept. of Economics, Northwestern University, USA

Received 24 April 2010; final version received 22 July 2011; accepted 1 September 2011
Available online 10 September 2011

Abstract
Orderings of interdependence are useful in many economic contexts: in assessing ex post inequality under
uncertainty; in comparing multidimensional inequality; in valuing portfolios of assets or insurance policies;
and in assessing systemic risk. We explore five orderings of interdependence for multivariate distributions:
greater weak association, the supermodular ordering, the convex-modular ordering, the dispersion ordering,and the concordance ordering. For two dimensions, all five are equivalent, whereas for three dimensions,the first four are strictly ranked and the last two are equivalent, and for four or more dimensions, all five arestrictly ranked. For the special case of binary random variables, we establish some equivalences among theorderings.
© 2011 Elsevier Inc. All rights reserved.
JEL classification: D63; D81; G11; G22
Keywords: Dependence ordering; Stochastic orders; Supermodularity; Weak association; Concordance✩

We thank Larry Epstein, Ian Jewitt, Paul Klemperer, and Jeremy Large, as well as seminar audiences at Oxford,
ESSET Gerzensee, and the Paris Tech-JET Conference on Inequality and Risk. We are very grateful to the referees andthe Associate Editors for their very helpful suggestions.
* Corresponding author.
E-mail addresses: margaret.meyer@nuffield.ox.ac.uk (M. Meyer), b-strulovici@northwestern.edu (B. Strulovici).
0022-0531/$ – see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.jet.2011.09.001

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1. Introduction
1.1. Motivation
In many economic contexts, it is of interest to know whether one set of random variables
displays a greater degree of interdependence than another. This paper explores several orderingsof interdependence for multivariate distributions and establishes the relationships among the orderings, both in general as well as in the important special case when all random variables arebinary.
Orderings of interdependence are applicable in several welfare-economic contexts. In manygroup settings where individual outcomes (e.g. rewards) are uncertain, members of the group may
be concerned, ex ante, about how unequal their ex post rewards will be (Meyer and Mookherjee [40]; Ben-Porath et al. [9]; Gajdos and Maurin [26]; Adler and Sanchirico [2]; Chew andSagi [17]). (This concern is distinct from concerns about the mean level of rewards and about
their riskiness.) Comparisons of reward schemes then require comparisons of the degree of interdependence of the random rewards.
Another welfare-economic application concerns comparisons of inequality when individuallevel data are available on multiple attributes such as income, health, and education (Atkinson and Bourguignon [6]; Bourguignon and Chakravarty [11]; Atkinson [5]). As long as thefunction aggregating the different attributes into an overall measure of welfare or deprivation for individuals is not additively separable across attributes, comparisons of multidimensional inequality will necessitate comparisons of the degree of interdependence among theattributes.1
Interdependence orderings are also useful for comparing the efficiency of matching mechanisms. In many matching contexts, perfectly assortative matching would be efficient and would
correspond to a “perfectly positively dependent” joint distribution of the random variables representing quality in each category (dimension). When, however, matches are formed based only on
noisy or coarse information (McAfee [38]), or when search is costly (Shimer and Smith [51]),or when signaling is constrained by borrowing constraints (Fernandez and Gali [25]), perfectly assortative matching will generally not arise. Fernandez and Gali [25] and Meyer andRothschild [41] apply bivariate dependence orderings to compare the performance of different
matching institutions.2
Orderings of interdependence can also be applied to compare the degree of alignment in thepreferences of members of decision-making groups (Boland and Proschan [10]; Baldiga andGreen [7]). Relatedly, in a strategic model of consensus building within a committee, Caillaud
and Tirole [13] study how the degree of interdependence of members’ ex ante uncertain payoffs
from a proposal affects the proposer’s optimal persuasion strategy.
In theoretical or experimental studies of information transmission in networks (CalvoArmengol and Jackson [14], Choi, Gale, and Kariv [19]), interdependence orderings can be used
1 There are important formal differences between these two welfare-economic applications. In the former applica-tion, the number of dimensions represents the number of individuals in the group, whereas in the latter, the number ofdimensions represents the number of attributes.
2 In a related application, Prat [47] explores how the composition of employee teams affects interdependence in (ex ante
random) decisions of team members and shows how properties of the production function translate into preferences overteam composition.

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to assess how the degree of concordance of individuals’ ex ante uncertain actions or outcomesvaries with time and with social structure, as represented by the network.
Finally, in finance and insurance, valuing portfolios of assets or insurance policies requires
assessing the degree of interdependence among asset returns or insurance claims (Müller andStoyan [46]; Denuit et al. [21]). Financial economists and macroeconomists are, moreover,increasingly interested in measures and comparisons of systemic risk in financial and economic systems that reflect the interdependence in the returns of different financial institutions
or sectors (Acharya [1]; Allen, Babus, and Carletti [3]; Beale et al. [8]; Hennessy and Lapan [30]).
For the special case of bivariate distributions, economists and statisticians have shown that
two intuitive concepts of greater interdependence are in fact equivalent. Suppose we are comparing the degree of interdependence between (Y1 , Y2 ) with that between (X1 , X2 ), wherefor each i, Yi and Xi have the same marginal distribution. The first concept is lower-orthantdominance. Lower-orthant dominance of Y over X requires that for all points in the support of the random vectors Y and X, the cumulative distribution function (cdf) of Y be atleast as large as the cdf of X: this captures the requirement that the components of Y are
more likely than those of X to both be “low” together, for any thresholds determining the
precise meaning of “low”.3 The second concept of greater interdependence is supermodular
dominance, which requires that Ew(Y ) be at least as large as Ew(X) for all objective functions w that are supermodular. Supermodularity (see Topkis [53] and Section 2.3) is a natural property of an objective function to capture a preference for interdependence, since its
arguments are complements, not substitutes: For a supermodular function (of two or more arguments), the effect of an increase in the value of any argument is larger, the larger are thevalues of the other arguments. It has been shown, in the economics literature by Levy andParoush [37], Epstein and Tanny [23], and Atkinson and Bourguignon [6] and in the statistics literature by Tchen [52], that for two-dimensional random vectors with identical marginals,
lower-orthant dominance of Y over X is equivalent to Ew(Y )  Ew(X) for all supermodular
functions w.
Economists have, nevertheless, made very little progress in the development of orderings forcomparing interdependence in multivariate, as opposed to bivariate, distributions. Given the wide
variety of applications for such orderings in both theoretical and empirical work, this is disappointing and may seem, at first glance, surprising. However, as we now argue, the n-dimensionalcase is substantially more difficult than the 2-dimensional case, for several reasons.
First, whereas positive and negative interdependence are “mirror images” of each other in two
dimensions, this symmetry breaks down for more than two dimensions. In two dimensions, forany plausible concept of positive dependence, if Y1 and Y2 are positively dependent, then −Y1
and Y2 are negatively dependent. Moreover, for Y1 and Y2 with identical uniform marginals on[0, 1], perfect positive dependence corresponds to Y2 = Y1 , while perfect negative dependence
corresponds to Y2 = 1 − Y1 . For more than two dimensions, however, there is in general nosimple way to convert a positively interdependent random vector (Y1 , Y2 , . . . , Yn ) into a negatively interdependent one. And even for {Yi }ni=1 with identical uniform marginals on [0, 1],
while Y1 = Y2 = · · · = Yn represents perfect positive dependence, there is no obvious definition

3 Given the assumption of identical marginal distributions, lower-orthant dominance is equivalent to upper-orthantdominance, which requires that the components of Y be more likely than those of X to both be “high” together.

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of perfect negative dependence.4 Since multivariate concepts of greater and lesser interdependence should be applicable to distributions displaying either positive or negative dependence, thelack of symmetry between positive and negative dependence in n > 2 dimensions complicatesthe development of orderings.5
Second, for n > 2 dimensions, there are more distinct notions of greater interdependence thanthere are for n = 2. Section 2 below presents five dependence orderings and shows that, whereas
for n = 2, all five are equivalent, for n = 3, four are strictly ranked and only two are equivalent,
and for n > 3, all five are strictly ranked. Thus, the selection of orderings of interdependence ismore complicated for multivariate distributions than for bivariate ones.
Finally, even for a given ordering of greater interdependence, determining whether two multivariate distributions can be ranked according to it may be more difficult than for bivariate
distributions. For n = 2, checking lower-orthant dominance is straightforward: it requires comparing the cdf’s at every point in the support. For n > 2, some of the dependence orderings westudy below can be implemented in an analogous fashion, by checking a given set of inequalities
for each point in the support. However, for others, there exists no set of criteria that can be applied pointwise; for implementing these orderings, more sophisticated techniques or algorithms
need to be developed.
1.2. Outline
Section 2 presents five orderings of greater interdependence for multivariate distributions.
Three of these have received some attention in the statistics and economics literatures, while twoof them are new.
We begin with an ordering inspired by the dependence concept of association, a concept proposed by Esary, Proschan, and Walkup [24] and widely used in statistics.6 Since the definitionof association cannot be meaningfully reversed to yield a concept of negative association, weturn to the concept of weak association (defined by Burton, Dabrowski, and Dehling [12]) and
use it to define the ordering we term greater weak association. A random vector (Y1 , . . . , Yn )displays greater weak association than a random vector (X1 , . . . , Xn ) if Y and X have identical marginals and, for all disjoint subsets A, B of {1, . . . , n} and nondecreasing functions r, s,
Cov(r(Yi , i ∈ A), s(Yj , j ∈ B))  Cov(r(Xi , i ∈ A), s(Xj , j ∈ B)).
We then present the (multivariate) supermodular ordering, which is the natural multivariate
generalization of the concept of supermodular dominance discussed above for bivariate dis4 For the set of all distribution functions with given marginals F , F , . . . , F , there exist upper and lower bounds forn
1 2

the distribution function, termed Fréchet bounds. The upper Fréchet bound is the natural candidate for the distribution
exhibiting maximal positive dependence. However, while the lower Fréchet bound might seem like a natural candidate
for the distribution displaying maximal negative dependence, the lower Fréchet bound is not in fact a proper distributionfunction except in very special cases (which do not include the uniform example described in the text). See Joe [36] andMüller and Stoyan [46] for more details.
5 Unfortunately, this lack of symmetry between positive and negative dependence for more than two dimensions is notalways recognized. See, for example, the discussion in Galeotti et al. [27, footnote 12].
6 Association, though a strong concept of dependence, is strictly weaker than affiliation, which is familiar to economists
from the work of Milgrom and Weber [44], who used it to formalize positive interdependence among bidders’ valuationsin auctions. De Castro [16] also notes that affiliation is strictly stronger than many other concepts of positive dependence.
Though association has not been widely used in economics, in a pair of recent papers Calvo-Armengol and Jackson [14],respectively [15] show that the employment statuses (respectively, wages) of individuals connected by a social network
are positively dependent in the sense of association.

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tributions. Meyer and Mookherjee [40] and Meyer [39] contain early proposals to use it as amultivariate dependence ordering, in the context of comparisons of ex post inequality under uncertainty. In the statistical literature, the multivariate supermodular ordering was formalized by
Shaked and Shanthikumar [49].
In Section 2.4, we introduce a new dependence ordering, the convex-modular ordering.
Greater interdependence for Y than for X according to the convex-modular ordering 
corresponds
to greater riskiness,
Stiglitz [48], of any aggregate ni=1 ri (Yi )n in the sense of Rothschild andcompared to i=1 ri (Xi ), for any choice of {ri }ni=1 nondecreasing. Convex-modular objectivefunctions are a natural way of capturing insurance companies’ preferences over the degree of
dependence in the claims they face.
Section 2.5 introduces another new dependence ordering, the dispersion ordering. The dispersion ordering is motivated by the idea that interdependence in random vectors can be comparedby comparing the dispersion of the cdf’s of their order statistics, with lower dispersion (assessed using the majorization ordering of vectors7 ) representing greater interdependence. We
show that greater interdependence for Y than for X according to the dispersion ordering corresponds
to greater riskiness, in the sense of Rothschild and Stiglitz [48], of the summary statistic
nni=1 I{Yi >si } compared toi=1 I{Xi >si } , for any point (s1 , . . . , sn ) in the support (where I is
the indicator function). A natural application of the dispersion ordering is to the assessment ofsystemic risk in financial systems, where systemic risk
n is greater the more “positively correlated”
are bank failures and where the summary statistic i=1 I{Yi >si } represents the number of banks
that would be solvent if bank i’s failure threshold were si .
Finally, we present the concordance ordering, which was formalized for multivariate distributions by Joe [35] and which combines the requirement of lower-orthant dominance with that ofupper-orthant dominance.8 The dispersion and concordance orderings share an important advantage relative to the other three orderings, namely that they are checkable pointwise.
Section 3 presents our main results, which establish the relationships among the five orderings
just described. Theorem 1 shows that for two dimensions, all five orderings are equivalent. Theorem 2, on the other hand, shows that i) for three dimensions, the first four orderings presented arestrictly ranked, and the dispersion and concordance orderings are equivalent, while ii) for four ormore dimensions, all five orderings are strictly ranked.
Section 4 focuses on binary random variables. Binary random variables, besides being common in theoretical, experimental, and empirical applications, also help to illuminate the structureof and relationships among the dependence orderings. We study a variety of special cases withbinary random variables, highlighting equivalences among the orderings in these cases. We offera brief conclusion in Section 5.
2. Orderings of greater interdependence
2.1. Preliminaries
We consider multivariate distributions with the same number, n, of variables and identical, finite support. Focusing on finite supports simplifies notation, avoids some uninteresting technicalissues, and clarifies the underlying structure of the orderings and the relationships between them.
7 See Hardy, Littlewood, and Pólya [29].
8 With more than two dimensions, even if Y and X have identical marginals, upper-orthant and lower-orthant domi-nance are no longer equivalent.

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Formally, let Li denote the finite, totally ordered set of values taken by the ith random variable, and let L denote the Cartesian product of the Li ’s. Li is a finite subset of R, and L
is a finite lattice of Rn with the following partial order: z  v if andnonly if zi  vi for alli ∈ N = {1, . . . , n}. If li denotes the cardinality of Li , then L has d = i=1 li elements.
As argued by Joe [36, p. 39], a natural desideratum for an ordering of interdependence isinvariance to monotonic relabelings of the elements of the support for each component. Formally,
we say that the ordering O is invariant to monotonic relabelings of coordinates if(Y1 , . . . , Yn ) O (X1 , . . . , Xn )




⇒
r1 (Y1 ), . . . , rn (Yn ) O r1 (X1 ), . . . , rn (Xn )(1)whenever ri : R → R, i ∈ {1, . . . , n}, are all nondecreasing. This invariance property captures
the idea that dependence orderings should be ordinal in that they should compare the extent towhich higher (respectively, lower) realizations in one dimension are accompanied by higher (respectively, lower) realizations in other dimensions, regardless of the cardinal scales used for eachdimension. An example of monotonic relabeling of coordinates is a componentwise coarseningof the support, in which some adjacent elements in the support of each component are combined. For example, starting with the lattice L = {0, 1, 2}n , the nondecreasing transformations ri
with, for all i, ri (0) = 0 and ri (1) = ri (2) = 1 coarsen the lattice to {0, 1}n by combining therealizations {1, 2} in each dimension.
All of the interdependence orderings we consider will be invariant to monotonic relabelingsof coordinates, as defined in (1). This is straightforward to confirm from their formal definitions
below. Consistent with our definition of invariance in (1), in the formal definitions of the orderings, when we consider monotonic functions, we allow these functions to be nondecreasing
rather than requiring them to be increasing. None of our results is affected by this choice, but
proofs are simplified by the ability to use nondecreasing indicator functions (denoted by I ).
While all of our interdependence orderings satisfy the invariance property in (1), some widelyused orderings, such as the (bivariate) linear correlation coefficient, fail to satisfy this invarianceand, a fortiori, are not invariant to a coarsening of the support. A higher value of the linear correlation coefficient for one bivariate empirical distribution than another is therefore not a robust indicator of greater dependence in the former distribution, since the ranking might be reversed by aggregation of the data. To see this, for L = {l, m, h}2 , where l < m < h, let (Y1 , Y2 ) have distribution g, where g(l, m) = g(m, l) = g(m, m) = g(h, h) = 14 , and let (X1 , X2 ) have distribution f ,where f (l, l) = f (m, m) = f (m, h) = f (h, m) = 14 . Then corr(Y1 , Y2 ) >(<) corr(X1 , X2 ) if
(l+h)
2 >(<) m. This in turn implies that if the support L is coarsened by combining the realizations l and m in each dimension, then corr(Y1 , Y2 ) > corr(X1 , X2 ), while if instead m and h are
combined, then corr(Y1 , Y2 ) < corr(X1 , X2 ).
Given that the orderings we consider below all satisfy the invariance property in (1), it is
without loss of generality, and convenient notationally, to assume henceforth that the sets Li ofvalues taken by the ith random variable each have the form {0, 1, . . . , li − 1}.
The lattice structure of the support L and its corresponding order are useful for comparing
distributions. One may label the d elements (or “nodes”) of L and view real functions on L as
vectors of Rd , where each coordinate of the vector corresponds to the value of the function at a
specific node of L. Similarly, a multivariate distribution whose support is L can be represented as
an element of the unit simplex d of Rd . For any function w : L → R and distribution f ∈ d ,the expected value of w given f is then the scalar product of w with f , seen as vectors of Rd :

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E[w|f ] =

w(z)f (z) = w · f,z∈Lwhere · denotes the scalar product of w and f in Rd .
Suppose the random vectors Y and X have distributions g and f , respectively, and the former
distribution dominates the latter according to some ordering “O”. We will use the phrases “Ydominates X according to the ordering O” and “g dominates f according to the ordering O”
interchangeably, and we denote the former by Y O X and the latter by g O f .
Given two distributions g and f on L, define the difference between them byδ ≡ g − f.

(2)

Most (though not all) of the orderings we consider are difference-based orderings, in that whether
or not two distributions g and f satisfy the ordering depends only on the corresponding δ. We
will frequently exploit this convenient property of difference-based orderings.9
2.2. Greater weak association
We begin with an ordering of greater interdependence that is inspired by the dependence
concept of association. Esary, Proschan, and Walkup [24] defined association as follows:
Definition 1 (Association). A random vector Y with support L is associated if for all nondecreasing functions r, s : L → R, Cov(r(Y ), s(Y ))  0.
A concept of negative association defined by reversing the inequality in the definition above
would be uninteresting: Y could only be negatively associated in this strong sense if it wereconstant (to see this, consider functions r = s). This is an important motivation for studying a lessstringent concept, weak association, defined by Burton, Dabrowski, and Dehling [12], along with
its negative counterpart, negative association, defined by Joag-Dev and Proschan [34]. In contrast
to association, which allows both r and s to depend on the entire vector Y , weak association andnegative association restrict them to depend on disjoint components of Y .10
Definition 2 (Weak association and negative association). A random vector Y with support L =ni=1 Li is weakly associated (respectively, negatively associated) if for any pair (A, B) of disjoint subsets of {1, . . . , n} and nondecreasing functions r : i∈A Li → R and s : j ∈B Lj → R,
Cov(r(Yi , i ∈ A), s(Yj , j ∈ B))  0 (respectively, Cov(r(Yi , i ∈ A), s(Yj , j ∈ B))  0).

×

×

×

We can now define an interdependence ordering corresponding to weak association and negative association as follows.11
Definition 3 (Greater weak association). Y displays greater weak association than X, denotedby Y GWA X, if they have identical univariate marginal distributions and for all disjoint subsets
9 Some of the orderings of interdependence considered in this paper can be modified so they are responsive not only

to the interdependence of the elements of a random vector but also to the levels of the elements. For clarity of focus, wewill not explicitly consider these modifications of the orderings or the relations between them.
10 It is clear from the definitions that association is a stronger concept than weak association. Hu, Müller, andScarsini [31] have shown that it is strictly stronger, even for two dimensions.
11 In the insurance literature, this ordering has been termed the correlation order—see Denuit et al. [21].

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×

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×

A, B of {1, . . . , n} and nondecreasing functions r : i∈A Li → R and s : j ∈B Lj → R,Cov r(Yi , i ∈ A), s(Yj , j ∈ B)  Cov r(Xi , i ∈ A), s(Xj , j ∈ B) .
Note that Definition 3 assumes that the random vectors being compared have identical univariate marginals; this is not an implication of the condition of greater covariance per se. In
Section 3, we show that for n  3, greater weak association is not a difference-based ordering,
though Theorem 1 in Section 3 implies that for the special case of n = 2, it is.12
The greater weak association ordering has the desirable feature that (Y1 , . . . , Yn ) are weakly
associated if and only if Y displays greater weak association than its independent counterpart,defined as the random vector X such that (X1 , . . . , Xn ) are independent and, for each i, Xi and Yi
have the same distribution.13 It might seem tempting to define Y as displaying “greater association” than X if Y and X have identical marginals and for all nondecreasing functions r ands defined on L, Cov(r(Y ), s(Y ))  Cov(r(X), s(X)). However, such a definition would havethe unappealing consequence that an associated random vector Y would not necessarily display
greater association than its independent counterpart.14
2.3. The supermodular ordering
For any z, v ∈ L, denote by z ∧ v the componentwise minimum of z and v, i.e., the element of L such that for each i, (z ∧ v)i = min{zi , vi } ∈ Li . Let z ∨ v similarly denote thecomponentwise maximum of z and v. A function w is said to be supermodular (on L) ifw(z ∧ v) + w(z ∨ v)  w(z) + w(v) for all z, v ∈ L. Denote by ei the unit vector in the ith dimension. Supermodular functions are characterized by the following property (see Topkis [53]):w(z + ei + ej ) + w(z)  w(z + ei ) + w(z + ej )(3)

for all i = j and z such that z + ei + ej ∈ L.15
Definition 4 (Supermodular ordering). Let the random vectors Y and X have distributions g
and f , respectively. The distribution g dominates the distribution f according to the supermodular ordering, written as g SPM f , if and only if E[w|g]  E[w|f ] for all supermodular
functions w.
It is clear from the definition that the supermodular ordering is a difference-based ordering.
12 To verify the claim for n = 2 directly, note that the only non-trivial partition of {1, 2} into disjoint subsets is {1}, {2}.

Then given Definition 3’s requirement that Y and X have identical marginals, Cov(r(Y1 ), s(Y2 )) − Cov(r(X1 ), s(X2 )) =E[r(Y1 )s(Y2 )] − E[r(X1 )s(X2 )], and the sign of the right-hand side depends only on the difference between the distributions of Y and X.
13 Similarly, (Y , . . . , Y ) are negatively associated if and only if the independent counterpart of Y displays greatern1
weak association than Y .
14 For the support {0, 1, 2}2 , let Pr(Y = Y = 0) = Pr(Y = Y = 1) = Pr(Y = Y = 2) = 1 and Pr(Y = 0,
1
2
1
2
1
2
1
4
1
Y2 = 2) = Pr(Y1 = 2, Y2 = 0) = 8 . Then it can be checked that Y is associated. However, consider the nondecreasing
functions r(Y1 , Y2 ) = I{Y1 1,Y2 1} and s(Y1 , Y2 ) = I{Y1 =2 or Y2 =2} . It is easy to confirm that Cov(r(Y1 , Y2 ), s(Y1 , Y2 ))is strictly smaller for Y than for its independent counterpart.
15 For functions w defined on Rn and twice differentiable, an equivalent characterization is: w is supermodular if and2

w  0 for all z ∈ Rn and all i = j .
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To see most clearly the appeal of the supermodular ordering as an ordering of greater interdependence, consider two distributions g and f such that, for some z ∈ L such that z + ei + ej ∈ L,
the difference δ ≡ g − f satisfies
δ(z) = δ(z + ei + ej ) = −δ(z + ei ) = −δ(z + ej ) = α(4)for some α > 0, and such that δ(v) = 0 for all other nodes v of L. In such a case, we say the
distribution g is obtained from f by an elementary transformation (ET) of size α on L whichleaves unchanged the probability of all nodes other than z, z + ei , z + ej , and z + ei + ej andwhich raises the probability of nodes z and z + ei + ej by the common amount α, while reducing
the probability of nodes z + ei and z + ej by the same amount. Intuitively, such ETs increase
the degree of interdependence of a multivariate distribution, as for some pair of components iand j , they make jointly high and jointly low realizations more likely, while making realizations
where one component is high and the other low less likely. Furthermore, they raise interdependence without altering the marginal distribution of any component. From (3), a function w issupermodular if and only if w · δ  0 for any δ of the form (4). Hence the class of supermodular
functions is precisely the class for which the expectation is raised by any ET as defined in (4).
Meyer and Strulovici [42] use duality methods to characterize the supermodular ordering anddevelop several constructive methods to implement this characterization. In their characterization, the elementary transformations defined above play a similar role to that of mean-preserving
spreads in Rothschild and Stiglitz [48] and Pigou–Dalton transfers in Atkinson [4] and Dasgupta,
Sen, and Starrett [20].16
If g SPM f , then g and f have identical univariate marginal distributions. To see this, notethat for any i ∈ {1, . . . , n} and k ∈ Li , the functions w(z) = I{zi k} and w(z) = I{zi <k} are bothsupermodular. Therefore g SPM f implies that, for each i ∈ {1, . . . , n} and k ∈ Li ,
0  E[w|g] − E[w|f ] =g(z) −
f (z) andz: zi k0  E[w|g] − E[w|f ] =z: zi <kz: zi kg(z) −

f (z),(5)

z: zi <kand these inequalities together imply that g and f have identical univariate marginal distributions.
2.3.1. The symmetric supermodular ordering
In many contexts, it is natural to assume that the supermodular objective functions being used
to compare distributions are symmetric with respect to the components of the random vectors. For
example, when the function w is an ex post welfare function defined on the realized utilities ofn individuals, as in the assessment of ex post inequality under uncertainty, it is natural to assume
that welfare is invariant to permutations of a given n-vector of utilities over the individuals. We
now formally define the symmetric supermodular ordering.
16 Elementary transformations of bivariate distributions were also used by Epstein and Tanny [23] and Tchen [52] toprove the equivalence, in two dimensions, of the supermodular and lower-orthant orderings for distributions with identicalmarginals. Our definition of ETs in the text is more restrictive, as it requires that the four points affected by the ET beadjacent points in the lattice; this more restrictive definition allows a much simpler proof of the two-dimensional result
and, more importantly, greatly facilitates the constructive methods for multivariate distributions developed in Meyer andStrulovici [42].

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×n

Call a lattice L = i=1 Li symmetric if Li = Lj for all i = j . Let θ denote a real function
on a symmetric lattice L. Depending on the context, θ can represent an objective function w ora probability distribution f . We will say that the function θ is symmetric on L if θ (z) = θ (σ (z))
for all z ∈ L and for all permutations σ (z) of z.
Definition 5 (Symmetric supermodular ordering). Let the random vectors Y and X have distributions g and f , respectively, on a symmetric lattice. The distribution g dominates the distribution
f according to the symmetric supermodular ordering, written as g SSPM f , if and only if
E[w|g]  E[w|f ] for all symmetric supermodular functions w.
For an arbitrary (not necessarily symmetric) function θ , the symmetrized version of θ , θ symm ,is defined as follows: for any z,
θ symm (z) =1  θ σ (z) ,n!

(6)σ ∈Σ(n)where Σ(n) is the set of all permutations of {1, . . . , n}. Importantly, if w is a supermodular
function, then w symm is supermodular. For a symmetric supermodular function w, let W symm (w)denote the set of supermodular functions ŵ on L such that the symmetrized version of ŵ is w,
i.e., ŵ symm = w. Note that {W symm (w)} is a partition of the set of all supermodular functions on
the symmetric lattice L. We can now state the following useful result:
Proposition 1. Given a pair of distributions g, f defined on a symmetric lattice L, g SSPM f ⇔g symm SPM f symm .
Proof. ⇒: If for all symmetric supermodular w, w · g  w · f , then for all symmetric supermodular w, w · g symm  w · f symm . In turn, if for some symmetric supermodular w, w · g symm 
w · f symm , then ŵ · g symm  ŵ · f symm for all ŵ ∈ W symm (w). Therefore, since {W symm (w)}w
partitions the set of all supermodular functions on L, w · g symm  w · f symm for all symmetricsupermodular w implies that g symm SPM f symm .
⇐: If for all supermodular w, w · g symm  w · f symm , then for all supermodular w,
w symm · g symm  w symm · f symm . This is equivalent to w symm · g symm  w symm · f symm for all symmetric supermodular w symm . This in turn implies that for all symmetric supermodular w symm ,w symm · g  w symm · f . 2
Proposition 1 states that one can characterize the symmetric supermodular ordering in termsof the supermodular order applied to symmetrized distributions.
The symmetric supermodular ordering has a very simple form for random vectors for which
each component has a binary support {0, 1}, so the lattice is L = {0, 1}n . To state the result, first
define, for a random vector Y with support L = {0, 1}n ,c(Y ) ≡n
i=1

I{Yi =1} .

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The “count function” c(Y ) gives the number of components of Y for which the realization takes
the value 1. Now recall the definition of the univariate convex ordering.17
Definition 6 (Univariate convex ordering). For random variables Z and V with support S ⊆ R,
Z dominates V according to the convex ordering, written as Z CX V , if Ew(Z)  Ew(V ) for
all convex functions w : S → R.
Since w(z) = z and w(z) = −z are both convex functions, Z CX V implies EZ = EV .
The convex ordering is equivalent to the ordering of greater riskiness studied by Rothschild andStiglitz [48].
Proposition 2. For random vectors Y and X distributed on L = {0, 1}n , Y SSPM X if and only
if c(Y ) CX c(X).
Proof. Any symmetric function w defined on L = {0, 1}n can be written asw(Y1 , . . . , Yn ) = φ c(Y1 , . . . , Yn ) ,(7)for some function φ : {0, 1, . . . , n} → R. Furthermore, a symmetric function w on {0, 1}n issupermodular if and only if the function φ(·) in (7) is convex. 2
In Sections 2.4 and 2.5, we propose two new orderings of greater interdependence.
2.4. The convex-modular ordering
In many contexts, the objective functions used to compare interdependence in multivariate
distributions have the formn
w(z) = φ r1 (z1 ) + · · · + rn (zn ) , with φ(·) convex and ri (·) i=1 nondecreasing.
We will term such functions convex-modular,
as they take a convex transformation of a modular(i.e. additively separable) aggregate, ni=1 ri (zi ). Any convex-modular function is supermodular,
and therefore the expectation of any convex-modular function is not decreased by any elementarytransformation of the form defined in (4). The proof of Proposition 2 rested on the observationnthat
n any symmetricn supermodular function w defined on L = {0, 1} is convex-modular, with
i=1 ri (zi ) =i=1 I{zi =1} . Convex-modular functions arise naturally in an insurance context,
where Z represents a vector of losses incurred by individuals 1, . . . , n, all of whom are insured bya given insurer, and where the insurance contract of individual i obliges the insurer to pay compensation ri (Zi ), which would take the form ri (Zi ) = min{mi , max{(1 − βi )(Zi − di ), 0}} for apolicy with a deductible di , a copayment rate βi for theninsured, and a compensation limit mi .
The total compensation paid out by the insurer is then i=1 ri (Zi ), and the insurer is concernedwith the riskiness of this total, so evaluates the cost of this payout using a convex objectivefunction φ.18 These observations motivate us to define the following difference-based ordering:
17 We do not require the support S to be convex, since we wish to apply the ordering to discrete random variables.

A function defined on S ⊂ R is convex if it can be extended to a convex function on R.
18 See Denuit et al. [21] for more details.

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Definition 7 (Convex-modular ordering). Let the random vectors Y and X have distributions g
and f , respectively. The distribution g dominates the distribution f according to the convexmodular ordering, written as g CXMOD f , if and only if E[w|g]  E[w|f ] for all convexmodular functions w.
This definition is equivalent to the requirement that E[w|g]  E[w|f ] for all functions w thatare nonnegative weighted sums of convex-modular functions.
definition that
n It follows fromthenn
Y CXMOD X if and only if, for all nondecreasing {ri }i=1 , i=1 ri (Yi ) CX i=1 ri (Xi ).
Since w(z) = I{zi k} and w(z) = −I{zi k} are both convex-modular functions, it follows,from the same logic as for the supermodular ordering, that Y CXMOD X implies that Y and X
have identical marginals.
It is natural to define a symmetric counterpart of the convex-modular ordering. It is clear fromour definition in (6) of the symmetrized version of a function, that if w is a convex-modular function, then wsymm is a nonnegative weighted sum of convex-modular functions. Let CM ∗ denotethe set of nonnegative weighted sums of convex-modular functions (where the dependence on agiven L is implicit). It follows from the above observation that CM ∗ is closed under symmetrization (although the set of convex-modular functions itself is not). As a consequence, we define thesymmetric convex-modular ordering as follows:
Definition 8 (Symmetric convex-modular ordering). Let the random vectors Y and X have
distributions g and f , respectively, on a symmetric lattice. The distribution g dominates the
distribution f according to the symmetric convex-modular ordering, written as g SCXMOD f ,if and only if E[w|g]  E[w|f ] for all symmetric functions w ∈ CM ∗ .
As noted above, g CXMOD f if and only if E[w|g]  E[w|f ] for all w ∈ CM ∗ . Using thisequivalence, it is then straightforward to adapt the proof of Proposition 1 to show:
Proposition 3. Given a pair of distributions g, f defined on a symmetric lattice L, g SCXMOD
f ⇔ g symm CXMOD f symm .
2.5. The dispersion ordering
Another notion of greater interdependence in Y than in X reflects the idea that the distributionfunctions of the order statistics of Y should be “closer together” or less dispersed than the distribution functions of the order statistics of X. To understand the link between dispersion of order
statistics and interdependence, suppose (Y1 , Y2 ) and (X1 , X2 ) both have symmetric distributions
on {0, 1}2 , and that Pr(Yi = 1) = Pr(Xi = 1) = 12 . Let Y1 and Y2 be perfectly positively dependent: the realizations (0, 0) and (1, 1) both have probability 12 . Let X1 and X2 be perfectly negatively dependent: the realizations (0, 1) and (1, 0) both have probability 12 . For the order statistics
of Y , min{Y1 , Y2 } and max{Y1 , Y2 }, Pr(min{Y1 , Y2 } = 0) = Pr(max{Y1 , Y2 } = 0) = 12 , so the twoorder statistics have the same distribution. By contrast, for those of X, Pr(min{X1 , X2 } = 0) = 1
while Pr(max{X1 , X2 } = 0) = 0, so the two order statistics have distributions as different as possible in this context. The qualitative lesson of this example is that for the more dependent random
vector Y , the distribution functions of the order statistics are more similar (less dispersed) thanfor X.
The majorization ordering of vectors can be used to formalize the notion of lower dispersion.
A vector a is said to be majorized by a vector b, written as a ≺ b, if i) the components of theAuthor's personal copy1472

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vectors have the same total sum, and ii) for all k, the sum of the k largest entries of a is weakly
smaller than the sum of the k largest entries of b (see Hardy, Littlewood, and Pólya [29]). Ifa ≺ b, then the components of a are less dispersed than the components of b.
Let Y(j ) denote the j th order statistic of Y , i.e. the j th smallest value from (Y1 , . . . , Yn ), anddefine X(j ) similarly. Let FY(j ) and FX(j ) denote the c.d.f.’s of these order statistics. For random
vectors with symmetric distributions, Shaked and Tong [50] suggested the following dependence
ordering.
Definition 9 (Symmetric dispersion ordering). For random vectors Y , X with symmetric distributions on a symmetric lattice L, the distribution of Y dominates that of X according to thesymmetric dispersion ordering, written as Y SDISP X, if the distribution functions of the order
statistics of Y are less dispersed than the distribution functions of the order statistics of X, that is, FY(1) (b0 ), . . . , FY(n) (b0 ) ≺ FX(1) (b0 ), . . . , FX(n) (b0 )∀b = (b0 , . . . , b0 ) ∈ L.

(8)

We now reformulate this dependence ordering in a manner which suggests a new ordering
which is both stronger and naturally applicable to asymmetric distributions. Each vector b =(b0 , . . . , b0 ) ∈ L can be seen as generating a (componentwise) binary coarsening of the supportof the random vector Y from L to {0, 1}n and a corresponding coarsened version of Y , Y b , suchthat Yib = 0 if Yi  b0 and Yib = 1 if Yi > b0 . Let Y(jb ) denote the j th order statistic of Y b , i.e. thej th smallest value from (Y1b , . . . , Ynb ) ∈ {0, 1}n , and let FY b denote the distribution function of(j )this order statistic. Then a condition equivalent to (8) is FY b (0), . . . , FY b (0) ≺ FXb (0), . . . , FXb (0)
(1)

(n)

(1)

(n)

∀b = (b0 , . . . , b0 ) ∈ L.

(9)

For asymmetric multivariate distributions, there is no particular reason to confine attention
to binary coarsenings generated by vectors with equal components b = (b0 , . . . , b0 ). We now
propose a new dependence ordering, which strengthens condition (9) by requiring that it hold forevery vector s = (s1 , s2 , . . . , sn ) ∈ L.
For a random vector Y , define its (componentwise) binary coarsening corresponding to thevector s ∈ L, Y s , byY s ≡ Y1s , . . . , Yns ∈ {0, 1}nwhere Yis =0 if Yi  si ,1 if Yi > si .

s , . . . , Y s ) ∈ {0, 1}n denote the vector of order statistics of Y s , that is, Y s equals the j th
Let (Y(1)(n)(j )smallest value from (Y1s , . . . , Yns ). Thus, Y(js ) = 0 if there are at least j values of i ∈ {1, . . . , n}such that Yi  si . Let FY(js ) denote the distribution function of Y(js ) .

Definition 10 (Dispersion ordering). For random vectors Y and X distributed on L, consider
the set of all binary coarsenings of Y and X, Y s and X s , respectively, corresponding to somes ∈ L. The distribution of Y dominates that of X according to the dispersion ordering, written
as Y DISP X, if for all s ∈ L, the distribution functions of the order statistics of Y s are lessdispersed than the distribution functions of the order statistics of X s , that is, s (0), . . . , FY s (0) ≺ FX s (0), . . . , FX s (0)
FY(1)
(n)
(1)
(n)

∀s ∈ L.

(10)

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The following proposition illuminates the appeal and convenience of the dispersion ordering19
by presenting some equivalent formulations. To state it, we define, for any s ∈ L,nn s sc Y =I{Yi =1} =I{Yi >si } ,
s

i=1

(11)

i=1

which counts the number of components of Y s that equal 1, or equivalently, the number of components of Y that strictly exceed the corresponding component of s.
Proposition 4. The following three conditions are equivalent:i) For Y, X with support L, Y DISP X.
ii) For all s ∈ L, Y s SSPM X s .
iii) For all s ∈ L, cs (Y s ) CX cs (X s ).
Proof. Since for any s, Y s and X s have support {0, 1}n , the equivalence of ii) and iii) follows
from Proposition 2 in Section 2.3.1. To show that ii) implies i), first note that since Y(js ) is the j thsmallest value from (Y1s , . . . , Yns ), FY(js ) (0) = Pr(Y(js ) = 0) = E[I{Y(js ) =0} ] is weakly decreasingin j . Hence the majorization condition (10) can be rewritten as
E

k

j =1

I{Y(js ) =0}  E

k

j =1

I{X(js ) =0}∀k ∈ {1, . . . , n}, ∀s ∈ L,

(12)with equality required for k = n. Now observe that
k

j =1

I{Y(js ) =0} =

k


k

I{cs (Y s )<n−(j −1)} =(1 − I{cs (Y s )n−(j −1)} )

j =1

j =1


  
= k − max cs Y s − (n − k), 0 .

(13)Since for any s ∈ L, cs (Y s ) is a symmetric function of Y s , and max{z − t, 0} is convex in z
for any t ∈ R, max{cs (Y s ) − (n − k), 0} is a symmetric supermodular function of Y s for allk ∈ {1, . . . , n}. Therefore, ii) and (13) imply that (12) holds for all k ∈ {1, . . . , n}. Setting k = n
in (13) gives
n

j =1

 
I{Y(js ) =0} = n − cs Y s ,(14)and since both cs (Y s ) and −cs (Y s ) are symmetric supermodular functions of Y s , ii) and (14)imply that for k = n, (12) holds with equality, as required. Thus ii) implies i).
19 It might, at first glance, seem more natural to define the dispersion ordering by requiring (10) to hold for all s innL̄ , where L̄i ≡ {−1, 0, 1, . . . , li − 1}. If si = −1, then Yis = 1 with probability 1, while
the extended lattice L̄ ≡i=1 i
if si = li − 1, then Yis = 0 with probability 1. Thus, including coarsenings corresponding to every s in L̄ would make×

the treatment of these two types of degenerate case symmetric. Nevertheless, it is straightforward to confirm that if (10)holds for all s ∈ L, then it holds for all s ∈ L̄, so the potential alternative definition is equivalent to the one used in thetext.

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To show that i) implies iii), first note20 that for random variables Z and V with support{0, 1, . . . , n},Z CX V⇔

E(Z) = E(V ) and∀t ∈ {1, . . . , n − 1},

E max{Z − t, 0}  E max{V − t, 0} .

(15)

Given (14), the equality in (12) for k = n implies that E(cs (Y s )) = E(cs (X s )). Given (13),the weak inequality in (12) for k ∈ {1, . . . , n − 1} implies that E[max{cs (Y s ) − t, 0}] E[max{cs (X s ) − t, 0}] for all t ∈ {1, . . . , n − 1}. Hence, for all s ∈ L, cs (Y s ) CX cs (X s ). 2
Proposition 4 provides a simple interpretation of the dispersion ordering for random vectors Yn
and X. Consider any coarsening of the support L = i=1 Li into {0, 1}n generated by, for eachdimension i, classifying all values zi  si as 0 and all values zi > si as 1, for some s ∈ L. Thencompare the distributions of the correspondingly coarsened random vectors Y s and X s according
to the symmetric supermodular ordering, or equivalently, given Proposition 2, use the univariate
convex ordering to compare the distributions of the random variables cs (Y s ) and cs (X s ) whichcount, respectively, the number of components of Y suchthat Xi > si .
that Yi > si and of X suchY DISP X if and only if, for any point s in the support, ni=1 I{Yi >si } is riskier than ni=1 I{Xi >si }in the sense of Rothschild and Stiglitz [48].
A natural application of the dispersion ordering is to the assessment of systemic risk in financial systems, where the n dimensions represent banks and where a bank’s financial health isoften summarized by classifying it as either solvent or insolvent. The degree of interdependence
in failures of financial institutions is of crucial importance because “as more banks fail in thesame time period, the economic disruption tends to increase disproportionately” (Beale et al. [8,p. 1]). This disproportionate increase in cost, representing an aversion to positive interdependence, can be captured by a systemic cost function φ(k), where k is the number of banks that failand φ is convex. It is natural to want rankings of expected systemic cost to be robust to variationsin the precise values of the failure thresholds for different banks. These considerations point towards using the dispersion ordering of the random vectors describing banks’ returns to compare
levels of systemic risk under different regulatory scenarios.
It is apparent from Proposition 4 that the dispersion ordering is a difference-based ordering.
Furthermore, if Y DISP X, then Y and X have identical univariate marginals. To see this, observethat (10) implies that for all s ∈ L,

×

n

j =1

F

Y(js )

(0) =

n

j =1

FX(js ) (0).

(16)

Now take s to be a vector with ith component equal to k ∈ Li and all other components j equal
to lj − 1. Then for any Y and X and for any j = i, Yjs = Xjs = 0. Hence the first n − 1 orders (0) = FX s (0). This in turnstatistics of both Y s and X s are 0, so it follows from (16) that FY(n)(n)
is equivalent to Pr(Yi  k) = Pr(Xi  k), and since this holds for all i and all k ∈ Li , Y and X
have identical marginals.
It follows from Proposition 4 that there is an easily described algorithm for checking whether
or not any given pair of n-dimensional random vectors Y and X satisfy Y DISP X: For eachs ∈ L, check whether or not cs (Y s ) CX cs (X s ). Since the support of cs (Y s ) and cs (X s ) is
20 See, for example, Jewitt [33] and Hardy, Littlewood, and Pólya [28].

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1475{0, 1, . . . , n}, checking cs (Y s ) CX cs (X s ) is equivalent, given (15), to checking n − 1 inequalities and 1 equality. Therefore, because an equality constraint can be rewritten as two inequality
constraints, to check Y DISP X, it is sufficient to check n + 1 inequalities for each s ∈ L.21 The
fact that Y DISP X is checkable pointwise is an important advantage of the dispersion ordering,
relative to the greater weak association, supermodular, and convex-modular orderings.
2.6. The concordance ordering
Another intuitively appealing notion of greater interdependence, the concordance ordering,has been formalized for multivariate distributions by Joe [35].
Definition 11 (Concordance ordering). Let the random vectors Y and X have distributions g
and f , respectively. The distribution g dominates the distribution f according to the concordance ordering, written as g CONC f , if and only if
Pr(Y  z)  Pr(X  z)

andPr(Y  z)  Pr(X  z)∀z ∈ L.

For any node in the support, the concordance ordering requires that the components of Y ,relative to those of X, be both more likely to be all higher than at that node and more likely to be
all lower than at that node.22
It is easy to see that the concordance ordering is a difference-based ordering. As is wellknown, Y CONC X implies that Y and X have identical univariate marginals.23 Note that theconcordance ordering shares with our dispersion ordering the desirable feature of being checkable pointwise.
3. Relationships among the orderings
We are now in a position to present our main results, which establish the relationships among
the orderings of interdependence defined in Section 2.
Theorem 1 (Orderings: two dimensions). For two dimensions, the following orderings are equivalent: greater weak association, supermodular ordering, convex-modular ordering, dispersion
ordering, and concordance ordering.
Proof. The equivalence for n = 2 between greater weak association, the supermodular ordering,and the concordance ordering is well known (see Meyer [39, Proposition 2] and Müller andStoyan [46, Theorem 3.8.2] for references). We now prove that for n = 2 the convex-modular
21 In practice, many of the inequalities generated as s varies will hold trivially or be redundant. For example, for

L = {0, 1}2 , the set of 12 inequalities that in principle need to be checked in fact reduces to only 5, corresponding to
the requirements that i) (Y1 , Y2 ) and (X1 , X2 ) have identical marginals (4 inequalities) and ii) Pr(Y1 = 0, Y2 = 0) 
Pr(X1 = 0, X2 = 0).
22 See Decancq [22] for a discussion of two classes of multivariate rearrangements which together underlie the concordance order.
23 To see this, first take z ∈ L to be a vector with ith component equal to k and all other components 0. Then Pr(Y  z) Pr(X  z) becomes Pr(Yi  k)  Pr(Xi  k). Similarly, taking z to be a vector with ith component equal to k − 1 andall other components equal to li − 1, Pr(Y  z)  Pr(X  z) becomes Pr(Yi  k − 1)  Pr(Xi  k − 1). Hence, for alli and k ∈ Li , Pr(Yi  k) = Pr(Xi  k), so Y and X have identical marginals.

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and dispersion orderings are equivalent to the other three orderings by showing that Y SPM
X ⇒ Y CXMOD X ⇒ Y DISP X ⇒ Y CONC X.
Since every convex-modular function is supermodular, the supermodular ordering implies theconvex-modular ordering. The fact that the convex-modular ordering implies the dispersion ordering follows from the proof of Proposition 4 and the facts that the functions max{cs (Y ) − (2 −k), 0}, for k ∈ {1, 2}, and −cs (Y ) are convex-modular. Finally, observe that by Proposition 4, if(Y1 , Y2 ) DISP (X1 , X2 ), then for all s ∈ L, Pr(cs (Y s ) = 0)  Pr(cs (X s ) = 0) and Pr(cs (Y s ) =2)  Pr(cs (X s ) = 2). The first of these inequalities is equivalent to Pr(Y1  s1 , aY2  s2 ) Pr(X1  s1 , X2  s2 ), while the second is equivalent to Pr(Y1 > s1 , Y2 > s2 )  Pr(X1 >s1 , X2 > s2 ). Hence (Y1 , Y2 ) DISP (X1 , X2 ) implies (Y1 , Y2 ) CONC (X1 , X2 ). 2
Theorem 2 (Orderings: three or more dimensions).
a) For n  3, greater weak association is strictly stronger than the supermodular ordering,which is strictly stronger than the convex-modular ordering, which is strictly stronger thanthe dispersion ordering, which is at least as strong as the concordance ordering.
b) For n = 3, the dispersion ordering is equivalent to the concordance ordering, whereas forn > 3, the dispersion ordering is strictly stronger than the concordance ordering.
Proof. i) The proof that Y GWA X implies Y SPM X is in Appendix A. Example 1 proves that
this implication is strict:
Example 1. Let
L = {0, 1}3 and let X and Y have distributions f and g, respectively. Let3
17 11 5 1
f (x) = 13 if
i=1 xi = 1 and f (x) = 0 otherwise. Let g(y) take the values 66 , 66 , 66 , 66
3when i=1 yi takes the values 0, 1, 2, 3, respectively. For these symmetric distributions, Propositions 1 and 2 together imply that g SPM f on L = {0, 1}3 if and only if c(Y ) CX c(X)on {0, 1, 2, 3}. From (15), c(Y ) CX c(X) on {0, 1, 2, 3} if and only if 1) Ec(Y ) = Ec(X); 2)
E[max{c(Y ) − 1, 0}]  E[max{c(X) − 1, 0}]; and 3) E[max{c(Y ) − 2, 0}]  E[max{c(X) −2, 0}]. It is easily checked that Ec(Y ) = Ec(X) = 1. Condition 2) holds since 17
66 = 2Pr(c(Y ) =3) + Pr(c(Y ) = 2)  2Pr(c(X) = 3) + Pr(c(X) = 2) = 0, and condition 3) holds since1
66 = Pr(c(Y ) = 3)  Pr(c(X) = 3) = 0. Hence, c(Y ) CX c(X) and g SPM f . However,11− 13 · 11< 0, while
for r(z1 ) = I{z1 =1} and s(z2 , z3 ) = I{z2 =z3 =1} , Cov(r(Y1 ), s(Y2 , Y3 )) = 66
Cov(r(X1 ), s(X2 , X3 )) = 0, so g GWA f does not hold.
ii) Since every convex-modular function is supermodular, Y SPM X implies Y CXMOD X.
We show that the implication is strict by providing, in Appendix B, an example of a supermodular
function on L = {0, 1, 2}3 which cannot be written as a nonnegative weighted sum of convexmodular functions.
iii) That Y CXMOD X implies Y DISP X follows from the proof of Proposition 4 and thefacts that the functions max{cs (Y ) − (n − k), 0}, for k ∈ {1, . . . , n}, and −cs (Y ) are all convexmodular. Example 2 proves that this implication is strict:
Example 2. Let L = {0, 1, 2} × {0, 1} × {0, 1}. Since both the convex-modular and the dispersion ordering are difference-based orderings, it is sufficient to specify δ ≡ g − f , the differ
ence between the distributions of Y and X. Let δ(z1 , z2 , z3 ) = > 0 if 3i=1 zi is even andδ(z1 , z2 , z3 ) = − < 0 if 3i=1 zi is odd. It is easily checked that Y CONC X. As proved inAuthor's personal copy
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1477

part iv) of the proof below, forn = 3, Y CONC X ⇔ Y DISP X. However, for the convexmodular function w(z) = max{( 3i=1 zi )−2, 0}, we have w ·(g −f ) = − < 0, so Y CXMOD X
does not hold.
iv) To show that Y DISP X implies Y CONC X, observe that, for k = 1, the left-hand sides =0} ] = 1 − Pr(Yi > si ∀i). Therefore, (12) implies that for allof (12) can be rewritten as E[I{Y(1)s ∈ L, Pr(Y > s)  Pr(X > s). Similarly, it follows from the equality in (12) for k = n and thes =0} ]  E[I{X s =0} ], which implies that for all
inequality for k = n − 1 that, for all s ∈ L, E[I{Y(n)(n)
s ∈ L, Pr(Y  s)  Pr(X  s). Hence Y CONC X.
Now we show that for n = 3, Y CONC X implies Y DISP X. First, for any s ∈ L, showingthat (12) holds with equality for k = n = 3 is equivalent, given (13) and (14), to showing that
E

3

i=1

I{Yi si } = E

3

I{Xi si } ,i=1

and this is true, since Y CONC X implies that Y and X have identical marginal distributions.
Given the equality in (12) for k = 3, it remains to show that for all s ∈ L,s =0} ]  E[I{X s =0} ]E[I{Y(1)(1)

s =0} ]  E[I{X s =0} ].
and E[I{Y(3)(3)

The first of these inequalities is equivalent to Pr(Y > s)  Pr(X > s) and the second to
Pr(Y  s)  Pr(X  s), as shown above in the proof that Y DISP X implies Y CONC X.
Therefore, both inequalities follow from Y CONC X.
Example 3 proves that for n > 3, Y CONC X does not imply Y DISP X:
Example 3. Let L = {0, 1}4 , let g, f represent the distributions of Y, X,
4and let δ(z1 , z2 , z3 , z4 ) ≡4g − f = > 0 if i=1 zi is even and δ(z1 , z2 , z3 , z4 ) = − < 0 if i=1 zi is odd. Again, it iseasily checked that Y CONC X. For s = (0, 0, 0, 0), consider the convex function of cs (z) defined by w = max{cs (z) − 2, 0}. We have w · (g − f ) = −2 < 0, so by Proposition 4, Y DISP X
does not hold. 2
Remark 1. Given the close relationship shown by Proposition 4 between the dispersion orderingand the symmetric supermodular ordering, and given that both are strictly weaker than the supermodular ordering, it is natural to ask how these two orderings relate to each other. For arbitrary
supports, neither implies the other; however, as we note in Section 4, in the special case of binary
random variables, the dispersion ordering is strictly stronger than the symmetric supermodular
ordering.
To confirm that for arbitrary supports, Y DISP X does not imply Y SSPM X, consider Example 2, used in part iii) of the proof of Theorem 2. Expand the support from L = {0, 1, 2} ×{0, 1} × {0, 1} to the symmetric L = {0, 1, 2}3 . For z ∈ L \ L, let δ(z) ≡ g − f = 0 and for z ∈ L,
define δ(z) as in Example 2. Then Y DISP X but, for the symmetric supermodular function w
defined in Example 2, w · δ = w · (g − f ) = − < 0, so Y SSPM X does not hold.
To confirm that Y SSPM X does not imply Y DISP X, consider Y, X with distributions g, f ,respectively,
on L = {0, 1}3 such that δ = g − f is given by δ(0, 0, 0) = 0, δ(z1 , z2 , z3 ) = > 0
3if i=1 zi = 1, δ(1, 1, 0) = −4 , δ(0, 1, 1) = δ(1, 0, 1) = − , and δ(1, 1, 1) = 3 . It is easy
to check that for k = 0, 1, 2, 3, Pr(c(Y ) = k) − Pr(c(X) = k) = 0, 3 , −6 , 3 , respectively, so
the distribution of c(Y ) is derived from the distribution of c(X) by a mean-preserving spread.
Therefore, c(Y ) CX c(X) and hence, by Proposition 2, Y SSPM X. However, P (Y1 = 1) −

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P (X1 = 1) < 0. Hence, Y and X do not have identical marginals, and therefore Y DISP X does
not hold.
Remark 2. For n  3, the strongest of our interdependence orderings, greater weak association,
is not a difference-based ordering. To show this, we extend Example 1, used in part i) of theproof of Theorem 2. For L = {0, 1}3 , define f and g as in Example 1. Now define f˜(x) = 12 if
3
3
1
˜
˜
x
=
0,
f
(x)
=
if
i
i=1
i=1 xi = 1, and f (x) = 0 otherwise, and let g̃(y) take the values
6

3
50
5 1
˜
i=1 yi takes the values 0, 1, 2, 3, respectively. By construction, g̃ − f =66 , 0, 66 , 66 wheng − f , and we showed in part i) of the proof of Theorem 2 that g GWA f does not hold. Yet(see Appendix C) g̃ GWA f˜. Hence for n  3, greater weak association is not difference-based.
Since the supermodular ordering is difference-based and the distributions in this example aresymmetric, it follows that even for symmetric distributions, greater weak association is strictly
stronger than the supermodular ordering whenever n  3.
4. Binary random variables
In many economic contexts, the random variables whose interdependence is to be assessed
are binary. Theoretical models often focus on binary action spaces or binary outcome spaces
for tractability. For example, Calvo-Armengol and Jackson’s [14] study of the effects of social
networks on interdependence in individuals’ employment outcomes suppressed wage variation
among employed workers and focused only on whether workers were employed or unemployed.
Experimental studies often focus on binary choice spaces to simplify the subjects’ decision problems as well as to simplify the data analysis. For example, Choi, Gale, and Kariv [19], in theirexperimental study of the effect of network structure on social learning and the resulting interdependence among agents’ decisions, focused on a decision environment with only two states of
the world, two signals, and two possible actions. In empirical work, for example on multidimensional inequality, binary classifications, such as whether or not income is below the poverty line
or whether or not an individual is literate, are often inevitable features of the data.
Binary random variables, besides being common, also help to illuminate the structure of andrelationships among the interdependence orderings. This section studies a variety of special caseswith binary random variables. Our aims are to highlight both i) equivalences among the orderings
that arise in these special cases and ii) easily checkable and easily interpretable necessary andsufficient conditions for the orderings to hold.
4.1. Symmetric distributions or symmetric objective functions
The following result is valid for any number n of dimensions.
Proposition 5.
a) For random vectors Y and X with symmetric distributions on L = {0, 1}n , the following
conditions are equivalent:i) Y SPM X;ii) Y CXMOD X;iii) Y DISP X;
iv) Y SDISP X;v) c(Y ) CX c(X).

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1479b) For random vectors Y and X distributed on L = {0, 1}n , the following conditions are equivalent:i) Y SSPM X;ii) Y SCXMOD X;iii) c(Y ) CX c(X).
Proof. a) It is clear from Definitions 9 and 10 that Y DISP X implies Y SDISP X. It follows
from Definition 9 and the proof of Proposition 4 that for symmetric distributions on L = {0, 1}n ,Y SDISP X if and only if c(Y ) CX c(X)—the only non-trivial choice of b = (b0 , . . . , b0 ) in
Definition 9 is (0, . . . , 0), in which case (8) is equivalent to c(Y ) CX c(X). But Propositions 1
and 2 together imply that for symmetric distributions on L = {0, 1}n , Y SPM X if and only ifc(Y ) CX c(X). The remaining equivalences then follow from Theorems 1 and 2.
b) Proposition 2 shows the equivalence of Y SSPM X and c(Y ) CX c(X) on L = {0, 1}n .
Since every symmetric w ∈ CM ∗ is supermodular, i) implies ii). Since c(Y ) is symmetric andconvex-modular, ii) implies iii). 2
For random vectors on L = {0, 1}n with asymmetric distributions, the dispersion ordering isstrictly stronger than the three orderings shown to be equivalent in part b) of Proposition 5. Withs = (0, 0, . . . , 0), the majorization condition (10) defining Y DISP X is equivalent to c(Y ) CX
c(X), but other choices of s generate additional conditions. In particular, as shown in Section 2.5,
Y DISP X implies that Y and X have identical marginals, whereas this is not an implication of
the three orderings in part b) of Proposition 5.24
4.2. Three dimensions
Proposition 6. For random vectors Y and X distributed on L = {0, 1}3 , the following conditions
are equivalent:i) Y SPM X;ii) Y CXMOD X;iii) Y DISP X;
iv) Y CONC X;v) c(Y ) CX c(X); Y and X have identical marginals; and for all i = j , Pr(Yi = Yj ) Pr(Xi = Xj ).
Proof. All of the orderings in the proposition are difference-based orderings, so it is sufficient
to work with δ ≡ g − f . For L = {0, 1}3 , the values of δ at each of the 8 nodes are displayed inFig. 1(a). Each of conditions i)–v) implies that Y and X have identical marginals (as shownin Section 2). This in turn implies that once the 4 values of δ(1, 1, 1) ≡ a, δ(0, 1, 1) ≡ b1 ,δ(1, 0, 1) ≡ b2 , and δ(1, 1, 0) ≡ b3 are specified (corresponding to the 4 nodes marked withblack dots in Fig. 1(a)), the remaining 4 values are determined, as specified in Fig. 1(a).
We first prove iv) and v) are equivalent. Given identical marginals, Y CONC X if and only if
3bi  0.
(17)
a  0,
a + bk  0 ∀k ∈ {1, 2, 3}, and 2a +i=1

24 The second example in Remark 1 following Theorem 2 is one for which L = {0, 1}3 and Y 
SSPM X, but Y and X
do not have identical marginals, so Y DISP X does not hold.

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(a)

(b)

Fig. 1. (a): δ = g − f for L = {0, 1}3 . (b): Labels for the nodes of L = {0, 1}3 .

From (15), c(Y ) CX c(X) on {0, 1, 2, 3} if and only if 1) Ec(Y ) = Ec(X); 2) E[max{c(Y ) −2, 0}]  E[max{c(X) − 2, 0}]; and 3) E[max{c(Y ) − 1, 0}]  E[max{c(X) − 1, 0}]. Given3 identical marginals, condition 1) holds, since (using Fig. 1(a)) Ec(Y ) − Ec(X) = 3a + 2 i=1 bi +(−3a − 2 3i=1 bi ) = 0. Conditions 2) and 3) are equivalent, respectively, to the first and third
inequalities in (17). Finally, the condition Pr(Yi = Yj )  Pr(Xi = Xj ) is equivalent, given
identical marginals, to Pr(Yi = Yj = 1)  Pr(Xi = Xj = 1), and this in turn is equivalent toa + bk  0, k = i, j . Hence, conditions iv) and v) are equivalent.
We now provide a simple constructive proof that for L = {0, 1}3 , Y CONC X implies
Y SPM X. Our constructive proof25 decomposes δ into 6 elementary transformations (ETs) of
the form defined in (4): for each of the 6 faces of the cube L = {0, 1}3 , there is one ET involving
the 4 nodes on that face. Using the labels in Fig. 1(b), let the ET involving nodes A, Bi , Bj , andCk have size α ij and the ET involving nodes Bk , Ci , Cj , and D have size α ij . It is easily checkedthat these 6 ETs sum to δ if and only if
a = α 12 + α 13 + α 23and ∀i, j, k ∈ {1, 2, 3}, i = j = k,a + bk = α ij + α ij .

(18)

Now seta + bkα ij = a
3a + 3i=1 biand α ij = 2a +

3

i=1


bi


a + bk.

3a + 3i=1 bi(19)

It is apparent by inspection that Eqs. (18) are satisfied and that, if the inequalities (17) defining
the concordance ordering hold, then for all i = j , α ij  0 and α ij  0. Thus, g CONC f implies the existence of a sequence of nonnegative ETs that sum to g − f . Since each ET raises
the expectation of any supermodular function, g SPM f . By Theorem 2, it then follows thatconditions i)–v) are all equivalent. 2
25 Hu, Xie, and Ruan [32, pp. 188–189] proved this in a very indirect manner using the tool of “majorization withrespect to weighted trees”.

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4.3. Four dimensions and “top-to-bottom” symmetry
The equivalence demonstrated in Proposition 6 for the three-dimensional cube between the
supermodular ordering and the concordance ordering breaks down if we increase either the number of dimensions, as shown by Example 3 (used in part iv) of the proof of Theorem 2), or thenumber of points in the support for some dimension, as shown by Example 2 (used in part iii)of the same proof). Nevertheless, some interesting equivalences do persist in higher dimensions
and can be demonstrated using a similar constructive method of proof.
Consider four-dimensional random vectors with support L = {0, 1}4 , and assume now that
their distributions satisfy a symmetry condition we term “top-to-bottom symmetry”. We say thatthe distribution of a random vector Z satisfies top-to-bottom symmetry if for any a ∈ {0, 1}4 ,P (Z = a) = P (Z = (1, 1, 1, 1) − a). Top-to-bottom symmetry arises naturally in a variety ofsettings. We give two examples: matching with frictions and social learning in networks.
In a matching context, suppose the four dimensions represent managers, supervisors, workers,
and firms, and suppose that for each dimension, there is one representative (individual or firm)with high quality (zi = 1) and one with low quality (zi = 0). Production requires forming a“team” consisting of exactly one manager, one supervisor, one worker, and one firm, and theoutput of such a team is a supermodular function of the qualities of each of its four components.
Supermodularity of the production function w implies that it would be output-maximizing for thefour high-quality individuals/firm to be matched and for the four low-quality individuals/firm to
be matched, in which case output would be w(1, 1, 1, 1) + w(0, 0, 0, 0). However, informational
frictions may prevent such an outcome being reached and cause the matching process to bestochastic. Nevertheless, as long as the stochastic process is certain to divide the individuals into
two teams, each consisting of one representative from each dimension, the distribution over team
quality vectors necessarily satisfies top-to-bottom symmetry. This is so since, if one of the teams
formed has individuals with qualities (a1 , a2 , a3 , a4 ) ∈ {0, 1}4 , the other team must consist of
individuals with qualities (1 − a1 , 1 − a2 , 1 − a3 , 1 − a4 ), given that for each dimension, there is
one representative with quality 0 and one with quality 1.
Choi, Gale, and Kariv [19] experimentally investigate the effect of network structure on the degree of interdependence of choices in social learning situations. Each agent i can choose between
two actions, 0 and 1, and the randomness in the environment generates, for each period, an ex ante
distribution over action vectors (a1 , . . . , an ) ∈ {0, 1}n . To simplify the computation of equilibrium behavior, they assume that the environment is symmetric with respect to the two states of
the world, the two signals, and the two actions. Given this symmetry, in each period and for any
network, the ex ante probabilities of the action vectors (a1 , . . . , an ) and (1 − a1 , . . . , 1 − an ) areequal, so the distributions satisfy top-to-bottom symmetry.
Proposition 7. For random vectors Y and X with distributions on L = {0, 1}4 that satisfy topto-bottom symmetry, the following conditions are equivalent:i) Y SPM X;ii) Y CXMOD X;iii) Y DISP X;
iv) c(Y ) CX c(X) and for all i = j , Pr(Yi = Yj )  Pr(Xi = Xj ).
The proof, in Appendix D, has a similar structure to that of Proposition 6. We use Proposition 4
to show that when L = {0, 1}4 and the distributions satisfy top-to-bottom symmetry, Y DISP X

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implies that iv) holds. (Top-to-bottom symmetry on L = {0, 1}4 itself ensures that Y and X haveidentical marginal distributions, since it implies that for all i, Pr(Yi = 1) = Pr(Yi = 0) = 12 andPr(Xi = 1) = Pr(Xi = 0) = 12 .) We then adapt the construction used to prove Proposition 6 toshow that the conditions in iv) ensure the existence of a sequence of nonnegative ETs that sum tog − f . Since each ET raises the expectation of any supermodular function, g SPM f and hence,by Theorem 2, conditions i)–iv) in Proposition 7 are all equivalent.
Even though, in the environment of Proposition 7, the SPM, CXMOD, and DISP orderings
are equivalent, we can show by example that the GWA ordering is strictly stronger and the concordance ordering strictly weaker.26
5. Conclusion
In this paper we examined five orderings of interdependence for n-dimensional distributions
and described a range of economic contexts in which these orderings are applicable, including
multidimensional inequality comparisons, assessments of ex post inequality under uncertainty,
valuations of portfolios of assets or insurance policies, and assessments of systemic risk in financial systems. While greater weak association, the supermodular ordering, and the concordanceordering have received some attention in the statistics and economics literatures, this paper introduces the dispersion ordering and the convex-modular ordering. Our dispersion ordering is
motivated by the link between greater interdependence in random vectors and lower dispersionof the cdf’s of their order statistics, and we proved that greater interdependence for Y than for X
according to the dispersion ordering corresponds
to greater riskiness,in the sense of Rothschildnand Stiglitz [48], of the summary statistic i=1 I{Yi >si } compared to ni=1 I{Xi >si } , for any point(s1 , . . . , sn ) in the support. Similarly, Y is more interdependent than X according
to the convexn nondecreasing, the aggregate n r (Y ) is riskier
modularordering
iffor
any
choice
of
{r
}
i i=1
i=1 i i
than ni=1 ri (Xi ). Theorem 1 showed that for n = 2, all five orderings of interdependence are
equivalent. In contrast, Theorem 2 showed that for n = 3, four of the orderings are strictly ranked,
and only the dispersion and concordance orderings are equivalent, while for n > 3, all five orderings are strictly ranked. For multivariate random vectors each of whose components are binary,
we demonstrated some equivalences among the orderings and provided easily checkable andeasily interpretable necessary and sufficient conditions for the orderings to hold. For arbitrary
multivariate distributions, we emphasized that the dispersion and concordance orderings havethe advantage, relative to the other three, of being checkable pointwise.
Appendix A. Proof that Y  GWA X implies Y  SPM X
This proof builds on Christofides and Vaggelatou’s [18] proof that if (Y1 , . . . , Yn ) is weakly
associated, then (Y1 , . . . , Yn ) dominates its independent counterpart according to the supermod26 To show that the GWA ordering is strictly stronger, let X have a uniform distribution on {0, 1}4 and let Y be ob1 ), one involving (0, 0, 1, 1), (1, 0, 1, 1), (0, 1, 1, 1),
tained from X by two ETs as defined in (4), each of size α ∈ (0, 16
and (1, 1, 1, 1) and the other involving (0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), and (1, 1, 0, 0). Both Y and X satisfy top-tobottom symmetry, and by construction, Y SPM X. However, for r(z1 , z2 , z3 ) = I{z1 +z2 +z3 2} and s(z4 ) = I{z4 =1} ,Cov(r(Y1 , Y2 , Y3 ), s(Y4 )) < 0 = Cov(r(X1 , X2 , X3 ), s(X4 )), so Y GWA X does not hold. To show that the concordance ordering is strictly weaker, recall Example 3, which was used in part iv) of the proof of Theorem 2. In thatexample, δ = g − f satisfies top-to-bottom symmetry, and as we showed, Y CONC X holds but Y DISP X does not.

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1483

ular ordering. First, we define a new random vector Z that has the same distribution as X and isindependent of Y . We then show by induction that Y SPM Z and hence Y SPM X.
For n = 2, the result is proved in the references cited in the proof of Theorem 1. Suppose that
it is true for n = m − 1, so that for all supermodular w, Ew(Y2 , . . . , Ym )  Ew(Z2 , . . . , Zm ).
Then
Ew(Z1 , Y2 , . . . , Ym ) =

l
i −1


E w(i, Y2 , . . . , Ym )Z1 = i Pr(Z1 = i)

i=0

=

l
i −1

E w(i, Y2 , . . . , Ym ) Pr(Z1 = i)(20)

E w(i, Y2 , . . . , Ym ) Pr(Y1 = i)

(21)

i=0

=

l
i −1
i=0



l
i −1

E w(i, Z2 , . . . , Zm ) Pr(Y1 = i)

i=0

=

l
i −1

E w(i, Z2 , . . . , Zm )|Y1 = i Pr(Y1 = i)

(22)

i=0

= Ew(Y1 , Z2 , . . . , Zm ).
The equality in (20) uses the independence of Z from Y , and that in (21) the assumptionthat Y and X, hence Y and Z, have identical marginal distributions (along with the fact thatE[w(i, Y2 , . . . , Ym )] is a univariate function of i). The inequality follows by applying the induction hypothesis for each value of i. The equality in (22) uses the independence of Z from Y .
Now we show that for all supermodular w,
Ew(Y1 , Y2 , . . . , Ym ) − Ew(Z1 , Z2 , . . . , Zm )
 Ew(Z1 , Y2 , . . . , Ym ) − Ew(Y1 , Z2 , . . . , Zm ),which combined with the previous steps yields the desired result. First observe that we can writew(Y1 , Y2 , . . . , Ym ) − w(Z1 , Y2 , . . . , Ym )
=

l
i −1



(I{Y1 >i} − I{Z1 >i} ) w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym ) .

i=0Hence,
Ew(Y1 , Y2 , . . . , Ym ) − Ew(Z1 , Y2 , . . . , Ym )
=

l
i −1





E I{Y1 >i} w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )

i=0



− E I{Z1 >i} w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )

=

l
i −1



i=0



E I{Y1 >i} w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )

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M. Meyer, B. Strulovici / Journal of Economic Theory 147 (2012) 1460–1489− E[I{Z1 >i} ]E w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )
=

l
i −1



− E[I{Y1 >i} ]E w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )
=

(23)



E I{Y1 >i} w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )

i=0

l
i −1





(24)



Cov I{Y1 >i} , w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )

i=0



l
i −1



Cov I{Z1 >i} , w(i + 1, Z2 , . . . , Zm ) − w(i, Z2 , . . . , Zm )i=0

= Ew(Z1 , Z2 , . . . , Zm ) − Ew(Y1 , Z2 , . . . , Zm ).

(25)

The equality in (23) uses the independence of Z from Y , and that in (24) the assumption that Y
and X, hence Y and Z, have identical marginal distributions. The inequality holds since i) I{Y1 >i}is a nondecreasing function of Y1 ; ii) (w(i + 1, Y2 , . . . , Ym ) − w(i, Y2 , . . . , Ym )) is a nondecreasing function of (Y2 , . . . , Ym ) since w is supermodular; and iii) by hypothesis, Y GWA X andhence Y GWA Z. The equality in (25) follows from the logic of the first four equalities. 2
Appendix B. Proof that for n  3, the supermodular ordering is strictly stronger than theconvex-modular ordering
Since every convex-modular function is supermodular, Y SPM X implies Y CXMOD X. We
show that the implication is strict by providing an example of a supermodular function on L ={0, 1, 2}3 which cannot be written as a nonnegative weighted sum of convex-modular ones:
Define w as follows: w(2, 2, 2) = 3, w(2, 2, 1) = w(2, 1, 2) = w(1, 2, 2) = 2, w(2, 2, 0) =w(2, 0, 2) = w(0, 2, 2) = 1, w(2, 1, 1) = w(1, 2, 1) = w(1, 1, 2) = 1, w(0, 2, 1) = 1, andw(z) = 0 for all other nodes z ∈ L. We first show that this function w(x) is not itself
convex-modular. Suppose
3 it i were. Then clearly the function φ(·) would have to take values in {0, 1, 2, 3}. If i=1 r (xi ) were strictly larger at (0, 2, 2) than at (0, 2, 1), then sincew(0, 2, 2) = w(0, 2, 1) = 1, φ(·) would
rise from 0 to 1 but then
3 noti be convex, since φ(·) would
remain constant at 1 even though i=1 r (xi ) increased. If, instead, 3i=1 r i (xi ) took on thesame value at (0, 2, 2) as at (0, 2, 1), then since 3i=1 r i (xi ) is modular (additively separable)in the xi ’s, it would follow that 3i=1 r i (xi ) took on the same value at (1, 2, 2) as at (1, 2, 1).
However, w(1, 2, 2) = 2 > 1 = w(1, 2, 1). Thus, we reach a contradiction, so w(z) as defined
above is not convex-modular.
In Meyer and Strulovici [42], we show how, for any finite support L, the “double description
method”, conceptualized by Motzkin et al. [45], can be used to determine the extreme rays ofthe cone of supermodular functions on L. For L = {0, 1, 2}3 , we show there that the function w
defined above is an extreme ray of the cone of supermodular functions and hence cannot be
non-trivially expressed as a nonnegative weighted sum of supermodular functions. This, combined with the fact that it is not itself convex-modular, shows that it cannot be expressed as anonnegative weighted sum of convex-modular functions. 2

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Fig. 2. δ(z) = g(z) − f (z) for L = {0, 1}4 and with “top-to-bottom symmetry” (which implies that c14 = c23 , c24 = c13 ,and c34 = c12 ). In the left cube, z4 = 0; in the right cube, z4 = 1.

Appendix C. Proof that in the example in Remark 2 (Section 3), g̃  GWA f˜
Since f˜ and g̃ are symmetric distributions, all partitions of {Z1 , Z2 , Z3 } into disjoint sets {Zi }and {Zj , Zk } will yield the same covariances. We therefore focus on the partition into {Z1 } and{Z2 , Z3 }. As is well known, any nondecreasing function on L = {0, 1}3 can be written as a nonnegative weighted combination of indicator functions, so it is sufficient to focus on functions r, sthat are nondecreasing indicator functions. The only non-trivial nondecreasing indicator functionof z1 is r(z1 ) = I{z1 =1} . There are four non-trivial nondecreasing indicator functions of (z2 , z3 ):s(z2 , z3 ) = I{z2 =1} , s(z2 , z3 ) = I{z3 =1} , s(z2 , z3 ) = I{z2 =z3 =1} , and s(z2 , z3 ) = I{z2 +z3 1} .
First take s(z2 , z3 ) = I{z2 =1} . Recall that g̃ − f˜ = g − f , where g and f are definedin Example 1. Therefore, since it was shown in Example 1 that g SPM f and since thesupermodular ordering is a difference-based order, g̃ SPM f˜. Note also that g̃ and f˜have identical marginal distributions, so Er(Y1 ) = Pr(Y1 = 1) = Pr(X1 = 1) = Er(X1 ) andEs(Y2 , Y3 ) = Pr(Y2 = 1) = Pr(X2 = 1) = Es(X2 , X3 ). Therefore, Cov(r(Y1 ), s(Y2 , Y3 )) −Cov(r(X1 ), s(X2 , X3 )) = E[r(Y1 )s(Y2 , Y3 )] − E[r(X1 )s(X2 , X3 )]. Since r(z1 )s(z2 , z3 ) =I{z1 =z2 =1} is supermodular, it thus follows from g̃ SPM f˜ that Cov(r(Y1 ), s(Y2 , Y3 )) 
Cov(r(X1 ), s(X2 , X3 )). An analogous argument holds if s(z2 , z3 ) = I{z3 =1} .
It remains only to consider s(z2 , z3 ) = I{z2 =z3 =1} and s(z2 , z3 ) = I{z2 +z3 1} . For s(z2 , z3 ) =I{z2 =z3 =1} , Cov(r(Y1 ), s(Y2 , Y3 )) − Cov(r(X1 ), s(X2 , X3 )) = 0, while for s(z2 , z3 ) = I{z2 +z3 1} ,Cov(r(Y1 ), s(Y2 , Y3 )) > 0 > Cov(r(X1 ), s(X2 , X3 )). Therefore, g̃ GWA f˜. 2
Appendix D. Proof of Proposition 7
Since the SPM, CXMOD, DISP, and CX orderings are all difference-based orderings, it issufficient to work with δ ≡ g − f . Fig. 2 displays the values of δ at each of the 16 nodes of
L = {0, 1}4 , given top-to-bottom symmetry of g and f , and hence of δ. Top-to-bottom symmetry of δ implies that once the 8 values of δ(1, 1, 1, 1) ≡ a, δ(0, 1, 1, 1) ≡ b1 , δ(1, 0, 1, 1) ≡ b2 ,δ(1, 1, 0, 1) ≡ b3 , δ(1, 1, 1, 0) ≡ b4 , δ(0, 1, 1, 0) ≡ c14 , δ(1, 0, 1, 0) ≡ c24 , and δ(1, 1, 0, 0) ≡ c34
are specified (corresponding to the 8 nodes marked with black dots in Fig. 2), the remaining
8 values are determined. Note that top-to-bottom symmetry implies that c12 = c34 , c13 = c24 ,Author's personal copy1486

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and c23 = c14 . Top-to-bottom symmetry of g and f also implies that Y and X have identical marginal distributions, since it implies that for all i, Pr(Yi = 1) = Pr(Yi = 0) = 12 andPr(Xi = 1) = Pr(X1 = 0) = 12 . From Fig. 2, identical marginals for Y and X here corresponds
to
a+

4


bh + cij + cik + cil = 0∀i = j = k = l.

(26)

h=1

We first show that condition iii) in the proposition implies condition iv). First note thatfrom Fig. 2, it is apparent that the second part of condition iv), Pr(Yi = Yj )  Pr(Xi = Xj )for all i = j , holds if and only if a + bi + bj + cij  0 for all i = j . Now use Proposition 4 to identify the implications of Y DISP X. For s = (0, 0, 0, 0), Yis = Yi and Xis = Xi
for all i, so cs (Y s ) CX cs (X s ) if and only if c(Y ) CX c(X). For s = (1, 1, 0, 0) and all permutations thereof, cs (Y s ) and cs (X s ) have support {0, 1, 2} and, from Eq. (15), cs (Y s ) CX
cs (X s ) implies that E[max{cs (Y s ) − 1, 0}] − E[max{cs (X s ) − 1, 0}]  0, which is equivalent toa + bi + bj + cij  0 for all i = j . Hence iii) implies iv).
In order to show that condition iv) implies condition i), we identify the inequalities towhich c(Y ) CX c(X) is equivalent here. From (15), c(Y ) CX c(X) if and only if 1) Ec(Y ) −Ec(X) = 0 and 2) E[max{c(Y ) − t, 0}] − E[max{c(X)− t, 0}]  0 for t = 1, 2, 3. Given4
4 topto-bottom symmetry, Ec(Y ) − Ec(X) = 4a + 3( h=1 bh ) + 2 · 2(cij + cik + cil ) + h=1 bh ,which equals 0 given4 (26). For t = 3, condition 2) corresponds to a  0, and for t = 2, it corresponds to 2a + h=1 bh  0. For t = 1 and given top-to-bottom symmetry, the left-hand side
of condition 2) is 3a + 2 4h=1 bh + 2(cij + cik + cil ), which equals a given (26). Therefore,c(Y ) CX c(X) here if and only if a  0 and 2a + 4h=1 bh  0.
We now show thata  0,

2a +4bh  0,and∀i = ja + bi + bj + cij  0⇒

g SPM f.

h=1

To do so, we decompose δ into 24 elementary transformations (ETs) of the form defined in (4):for each of the 24 faces of the hypercube L = {0, 1}4 , there is one ET involving the 4 nodes onthat face. We will abuse notation slightly and let the value of δ at a given node also serve as thelabel for that node. Let the two ETs involving the nodes a, bi , bj , and cij (there are two such ETs
because of the top-to-bottom symmetry of δ) have size βij = βj i . Let the two ETs involving thenodes bi , cik , cil , bj (once again, there are two such ETs because of the top-to-bottom symmetry)
have size αij = αj i . There are 6 distinct values of βij and 6 distinct values of αij . For the 24 ETs
so defined to sum to δ, it is necessary and sufficient that βij and αij satisfy27βij = a,i<jbi + βij + βik + βil = αij + αik + αil∀i = j = k = l,−cij + βij + βkl = αik + αil + αj k + αj l∀i = j = k = l.

(27)

27 Further details of this and subsequent arguments are provided in the supplementary material, Meyer andStrulovici [43], available on the journal website.

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These three (sets of) equations ensure that each of the (sets of) nodes labeled a, bi , and cij ,respectively, in Fig. 2 is transformed from its values under the distribution f to its values under
the distribution g by the sequence of ETs just defined. Eqs. (27) can be rearranged toβij = a,i<jbi + βij + βik + βil = αij + αik + αil
2αij + 2βkl = a + bi + bj + cij∀i = j = k = l,∀i = j = k = l.

(28)

Noting the similarity between Eqs. (28) and (18) for the 3-dimensional cube, seta(a + bi + bj + cij ), ∀i = j = k = l,
4a + 4h=1 bh
(2a + 4h=1 bh )(a + bi + bj + cij )αij =, ∀i = j.
2(4a + 4h=1 bh )
βkl =

(29)

Recalling that top-to-bottom symmetry of δ implies that c12 = c34 , c13 = c24 , and c23 = c14 andusing (26), it can be checked, with some algebra, that with these choices for βkl and αij , Eqs. (28)are satisfied. Furthermore, if a  0, 2a + 4h=1 bh  0 and for all i = j , a + bi + bj + cij  0,then βkl and αij as defined in (29) are all nonnegative. Thus, condition iv) in the proposition
implies the existence of a sequence of nonnegative ETs that sum to δ = g − f . Since each ET
raises the expectation of any supermodular function, g SPM f . By Theorem 2, it then follows
that conditions i)–iv) are all equivalent. 2
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